567

Journal of Oceanography, Vol. 56, pp. 567 to 581, 2000

Keywords:⋅ Ocean tide,⋅ loading tide,⋅ satellite altimetry,⋅ assimilation,⋅ tidal energydissipation.

* Corresponding author. E-mail: matumoto@miz.nao.ac.jp

Copyright © The Oceanographic Society of Japan.

Ocean Tide Models Developed by Assimilating TOPEX/POSEIDON Altimeter Data into Hydrodynamical Model:A Global Model and a Regional Model around Japan

KOJI MATSUMOTO1*, TAKASHI TAKANEZAWA2 and MASATSUGU OOE1

1Division of Earth Rotation, National Astronomical Observatory, Mizusawa 023-0861, Japan2The Graduate University for Advanced Studies, Mizusawa 023-0861, Japan

(Received 11 January 2000; in revised form 25 April 2000; accepted 17 May 2000)

A global ocean tide model (NAO.99b model) representing major 16 constituents witha spatial resolution of 0.5° has been estimated by assimilating about 5 years of TOPEX/POSEIDON altimeter data into barotropic hydrodynamical model. The new solutionis characterized by reduced errors in shallow waters compared to the other two mod-els recently developed; CSR4.0 model (improved version of Eanes and Bettadpur,1994) and GOT99.2b model (Ray, 1999), which are demonstrated in comparison withtide gauge data and collinear residual reduction test. This property mainly benefitsfrom fine-scale along-track tidal analysis of TOPEX/POSEIDON data. A high-resolu-tion (1/12°) regional ocean tide model around Japan (NAO.99Jb model) by assimilat-ing both TOPEX/POSEIDON data and 219 coastal tide gauge data is also developed.A comparison with 80 independent coastal tide gauge data shows the better perform-ance of NAO.99Jb model in the coastal region compared with the other global mod-els. Tidal dissipation around Japan has been investigated for M2 and K1 constituentsby using NAO.99Jb model. The result suggests that the tidal energy is mainly dissi-pated by bottom friction in localized area in shallow seas; the M2 ocean tidal energyis mainly dissipated in the Yellow Sea and the East China Sea at the mean rate of 155GW, while the K1 energy is mainly dissipated in the Sea of Okhotsk at the mean rateof 89 GW. TOPEX/POSEIDON data, however, detects broadly distributed surfacemanifestation of M2 internal tide, which observationally suggests that the tidal en-ergy is also dissipated by the energy conversion into baroclinic tide.

shallow seas (ocean depth < 1000 m); the RMS differ-ences is 9.78 cm for M2. This result suggests that largeambiguities in recent ocean tide models still remain inshallow water region. We discuss the origins of errors inthe tide models and aim at improving the ocean tide mod-els in the shallow waters.

The modern global tide models can be categorizedinto three groups; (1) empirical model, (2) hydrodynami-cal model, and (3) assimilation model. All of the recentocean tide models with each category have shortcomingsas to the shallow water tide modelings.

The empirical model is based on only observationaldata, e.g., Eanes and Bettadpur (1994), Desai and Wahr(1995), and Ray (1999). Since the common data sourceof these models is T/P, the resulting resolution of modelsis limited by the ground track spacings of the satellite,and the model domain is also necessarily limited by thedata coverage of the satellite (e.g., ±66° of latitude forT/P). The problem of coarse ground track spacings (2.83°

1. IntroductionSince the launch of satellite altimeter TOPEX/

POSEIDON (T/P), thanks to its unprecedented precise seasurface height data, the accuracy of ocean tide model hassignificantly improved especially in open ocean. The ac-curacy assessment by Andersen et al. (1995) suggestedthat the agreements of T/P-derived ocean tide models withopen-ocean tide gauge data are better than those of theolder Schwiderski (1980) and Cartwright and Ray (1991)models. Shum et al. (1997) indicated that the differencesamong eight T/P-based models are relatively small in deepocean (ocean depth > 1000 m); for instance, the RMSdifferences for M2 constituent is 0.97 cm. However, theypointed out that quite large differences occur mainly in

568 K. Matsumoto et al.

or 314 km at equatorial region for T/P) becomes criticalin shallow waters where the ocean tides varies rapidly inspace.

One can design the resolution of a hydrodynamicalmodel as high as desired, depending upon the computercapacity. However, numerical hydrodynamical tide mod-els always meet difficulties caused by the inaccuraciesarising from inadequate bathymetric data and unknownfrictional and viscosity parameters (Ray et al., 1996)which especially affect the modeling of shallow watertides. Kantha (1995) also pointed out that it is difficult toobtain the tide model accurately in particular its phasemodel without extensive fine-tuning of control param-eters such as the bottom friction. Although FES.94.1model (Le Provost et al., 1994) utilized finite elementmeshes and it has a very high resolution in shallow waterregions, its accuracy in such regions is still questionable(see Subsection 5.2).

We believe that the assimilation method is the mostpromising one resolving the problem in shallow waterregions. The problem of the empirical method with re-gard to the resolution can be compensated by the hydro-dynamical model, and the inadequate potential of thehydrodynamical model can be recovered by observationaldata. Other advantages of the assimilation model over the

empirical model are that (i) a truly global ocean tide modelcan be developed by extrapolating the model beyond thearea of latitudes 66° (N or S) which is the limit of T/Pground tracks, and that (ii) erroneous altimetric tidal so-lutions contaminated by non-tidal ocean variability canbe corrected for, in other words, the hydrodynamics isexpected to act as an effective data filter. However, manyinvestigators of assimilative models have not paid atten-tion so far to the nature of shallow water tides, that is, thepredominant short-wavelength components in shallowwaters. Le Provost et al. (1995) corrected for the long-wavelength error only by altimetric tides in their previ-ous FES.94.1 model. Egbert et al. (1994) and Matsumotoet al. (1995) constrained the hydrodynamic model bycoarsely distributed crossover data. Kantha (1995) appliedDesai and Wahr (1995) model to constrain only the deepregion of his model.

Han et al. (2000) assimilated T/P-derived along-trackM2 solutions into three-dimensional hydrodynamicalmodel off Newfoundland. They discussed about the ad-vantage of an assimilation model for ocean tides in shal-low water region. We basically follow this approach whichis characterized by making the altimetric tidal estimationwithin small bins along the T/P ground tracks, in order toconserve the short-wavelength characteristics of shallow

Table 1. Summary of the tidal modeling schemes, in which EMP = empirical model; ASS = assimilation model; BGM = back-ground model; AM = analysis method; RES = response method; HAR = harmonic method; FCN = FCN resonance effect isincluded in the response analysis or not; RAD = radiational potential is included in the response analysis or not; AS = assimi-lation scheme; REP = representer method of Egbert et al. (1994); NUD = nudging; OLI = optimal linear interpolation; AB =altimetric boundary condition; ATS = along-track tidal solutions; EE = estimation error of altimetric solution is taken intoaccount in assimilation or not; SAL = self-attraction/loading effect is taken into account or not; SAL_LA = linear approxima-tion of SAL; SAL_SCH = SAL computed from Schwiderski (1980) model; SAL_NAO.99b = SAL computed from NAO.99bmodel (this study).

Ocean Tide Models from TOPEX/POSEIDON 569

water tides. We counted in the free core nutation reso-nance effect and solar radiational effect for the better tidalanalysis. The empirical tidal solution based on observa-tion is used as an important constraint in the hydrody-namical model in which self-attraction/loading effect isprecisely estimated. Here, the hydrodynamical model isused as an interpolation device to develop a merged oceantide model. Table 1 summarizes modeling scheme of sevenselected models including this study to make the meth-odology of this study more clear. We develop a globalocean tide model (NAO.99b model) and a regional oceantide model around Japan (NAO.99Jb model) for 16 ma-jor constituents, i.e., M2, S2, N2, K2, 2N2, µ2, ν2, L2, T2,K1, O1, P1, Q1, M1, OO1, and J1.

2. TOPEX/POSEIDON Data ProcessingWe processed about five years of the MGDRB

(Merged Geophysical Data Records generation B) of thecycles 9-198. Our data processing procedure consists ofthe following five steps; (1) Unreliable data are elimi-nated by using the flag information contained in theMGDRB. The rejection criterion described in USER’SHANDBOOK Section 3.4 (Benada, 1997) was applied.However, we totally ignored ocean tide model flags(GEO_BAD_2 bits 1–4). (2) The 1 Hz MGDRB data areinterpolated to geographically fixed normal points in or-der to prevent the geoid gradients from contaminating seasurface height. (3) The standard geophysical correctionexcluding ocean tidal correction is applied to sea surfaceheight to form residual sea surface height ζ. (4) The domi-nant short-wavelength spatial variations of non-tidal sig-nal are removed from ζ by an along-track low-pass filter-ing by using a Gaussian filter with e-folding scale of 1°.However, the application of the low-pass filtering is lim-ited to deep seas (depth > 1000 m), because wavelengthof ocean tide might be short in the order of 100 km orless in shallow waters. (5) The filtered (in deep seas) orunfiltered (in shallow seas) residual height is averagedover small bins which are gridded along the ground tracks.The grid sizes are 0.5° for global model and 1/12° forregional model, which are the same as the spatial resolu-tion chosen for the hydrodynamical model. By cycle-by-cycle processing, we obtained the time series of residualsea surface height in each grid.

3. Tidal AnalysisThe residual sea surface heights are analyzed within

each grid using the response method (Munk andCartwright, 1966). The notable feature of the responsemethod is that the method does not insist upon express-ing the tides as sums of harmonic functions of specifiedtidal spectral line, but expressing the tides by smoothadmittance functions of each tidal species. Advantagesof the response method over standard harmonic decom-

position method in altimetric along-track tidal analysisis described by Ray (1998b). Observation equation usingthe response method with the orthotide extension byGroves and Reynolds (1975) is represented as

ζ t U P t V Q t Cml ml ml mll

l

m

m

( ) = ( ) + ( )[ ] + ( )==∑∑

00

2

1

where Pml(t) and Qml(t) are the orthotide functions whichare formed by forcing equilibrium tides, Uml and Vml arethe unknown parameters to be solved for, and C is alsounknown parameter which represents the offset in ζ dueto mean sea surface error, mean non-tidal sea surfacedynamic topography, and geographically correlated orbiterror. We set l0 = 0 and l1 = l2 = 2. l0 = 0 means that weassigned a constant admittance for the long period tidesin which we included annual (Sa) and semi-annual (Ssa)tides.

The standard response-orthotide method is slightlymodified to include the free core nutation (FCN) reso-nance effect and the radiational potential. Since the FCNeigenfrequency (about 1 + 1/430 cycle per sidereal day)is well within the diurnal tidal band, there will be a cor-responding resonance in the Earth’s response to the diur-nal tidal force, and hence the forced nutations and bodytide should be resonant at this frequency (Wahr and Sasao,1981). The effect of the FCN on ocean tide can be takeninto account by replacing the amplitudes of diurnal equi-librium tides with those multiplied by the following fac-tor,

Rk h

k hjj jω

ω ω21

2 21 2 21

2 2

1

12( ) =

+ ( ) − ( )+ −

( )˜ ˜

where k2 and h2 are nominal Love numbers, k2 and h2are frequency-dependent Love numbers of Wahr (1981)depending on FCN frequency for which 1 + 1/433.2 cy-cles per sidereal day (Herring et al., 1986) is applied.

The solar radiation indirectly affects S2 tide and itselliptical satellite T2 tide through atmospheric tides.Cartwright and Ray (1994) suggested that this atmos-pheric loading effect can be taken into account by multi-plying the amplitude of tide generating potential of S2and T2 by 0.97 and lagging their phases by 5.9°. This ad-justment is also applied to our analysis. We expect thatcross-talk between S2 and K2 are reduced and K2 con-stituent is improved (see Cartwright and Ray (1994) fordetails).

Uml, Vml and C are determined by the least squaresmethod. Estimation errors em for each species m are si-multaneously calculated in solving equation (1). They arethen normalized by the averaged value of the estimation

570 K. Matsumoto et al.

errors over all the estimation grids ( em = em/∑em), whichare used to define the modification factor in Eq. (13) (seeSubsection 4.3). The histogram of em (not shown) is bi-modal. About 69% of the T/P grids have value of em =1.0 ± 0.1, and 19% of them have value of em = 0.7 ± 0.1.The latter mainly consists of the grids in which cross-over points are contained, i.e., more data are availablecompared to the standard single-track grids. The ampli-tude and the Greenwich phase of 16 major constituentsare directly calculated from Uml and Vml. These altimeter-derived tidal solutions are hereinafter referred to as T/Psolutions.

4. Hydrodynamical Model and Assimilation

4.1 Hydrodynamical modelThe hydrodynamical model used in this study is

based on the tidal equations derived by Schwiderski(1980), which is a two-dimensional depth-integrated shal-low water equations. We improved them by estimatingsecondary loading effect more precisely, and by addingadvection term. It is appropriate to express the momen-tum equations and the continuity equation in sphericalpolar coordinate (R, λ , θ) = (Earth’s mean radius(=6.371 × 106 m), east longitude, colatitude), because thecurvature of the Earth can not be neglected for a globalcalculation of ocean tides. These equations are

∂∂

+ = ∂∂

+ −( ) − + +

( )

U

tA

gH

RV F Fo

b eλ λ λθ λ

γ η ξ ζ θsin

˜ cos2 2

3

Ω

∂∂

+ = ∂∂

+ −( ) + + +

( )

V

tA

gH

RU F Fo

b eθ θ θθ

γ η ξ ζ θ˜ cos2 2

4

Ω

∂∂

= − ∂( )∂

+ ∂∂

( )ζθ

θθ λ

o

t R

V U15

sin

sin

where η, ξ and ζo are forcing equilibrium tide, self-at-traction/loading (hereinafter referred to as SAL) term in-duced by ocean tide, and ocean tidal elevation to bemodeled with respect to ocean bottom, respectively; g(=9.81 ms–2) the gravity acceleration; Ω (=7.2722 × 10–5

radian s–1) the Earth’s angular velocity; H the local oceandepth which is taken from ETOPO5; γ 2 = 1 + k2 – h2 ; Uand V the east and south velocities which are integratedfrom the surface to the bottom of the ocean. Aλ and Aθ areadvection terms, which are retained only for the regionaltide model around Japan, being expressed as

AR

UV H UVH

HR

R

U H U H

H

UV

HR

λθ

λ

θ

θ λθ

= ∂( )∂

−

+∂( )

∂−

+ ( )

1

1 26

2 2

/

sin

/ cot

AR

V H V H

HR

R

UV H UVH

H

V U

HR

θθ

λ

θ

θ λθ

=∂( )

∂−

+ ∂( )∂

−

+ − ( )

1

17

2 2

2 2

/

sin

/ cot

where

HH

HH

H

Hθ λθ λ

= ∂∂

= ∂∂

1 1 and .

The bottom friction terms (Fλb and Fθ

b) are written as

FBU

HU Vb

λ = − +( ) ( )22 2 1 2

8/

FBV

HU Vb

θ = − +( ) ( )22 2 1 2

9/

where the bottom friction coefficient is defined as B =0.0026 × µsinθ, µ being the grading parameter ofSchwiderski (1980). The eddy viscosity terms (Fλ

e andFθ

e) are also fully described by Schwiderski (1980).In many numerical tidal models, the secondary SAL

term ξ is approximated by linear relationship with localocean tidal height as ξ = βζo, where values of 0.05 to0.12 are empirically used for constant β. However,Matsumoto (1997) and Ray (1998a) pointed out that thescalar approximation yields large error near coast andthere is no appropriate global constant generally chosenfor β. It is required for accurate tidal modeling in par-ticular to shallow waters that ξ be calculated by globalintegrals of the tidal elevation as (e.g., Hendershott, 1972)

ξ θ λ ρρ

ζ θ λ, ,max

( ) =+

⋅ ⋅ + −( ) ⋅ ( ) ( )( )=

∑ 3

2 11 10

0 nk hn n o n

n

n

′ ′

where ρ (=1035 kg m–3) and ρ (=5516 kg m–3) are themean density of sea water and the Earth, respectively; kn′and hn′ are potential and deformational load Love num-bers, respectively; ζo(n) is n-th degree spherical harmonicof tidal elevation as

Ocean Tide Models from TOPEX/POSEIDON 571

ζ θ λ ζ θ λo o nn

n

, ,max

( ) = ( ) ( )( )=

∑0

11

where we decomposed tidal elevation field up to degreenmax = 360.

For numerical computations, discrete equations aredeveloped corresponding to the continuous equations (3),(4), and (5), with 0.5° × 0.5° grid system for the globalmodel and 1/12° × 1/12° grid system for the regionalmodel around Japan. The latter has the model domain of(110°E–165°E, 20°N–65°N). We design a fully staggeredfinite-difference scheme following Schwiderski (1980).The computational northern limit for the global modelwas set at θ = 3° to avoid the singular point problem inthe north pole. FES.95.2 model (Le Provost et al., 1995)was used to provide boundary condition for the grids ofθ ≤ 3°. The mathematical boundary conditions adoptedare no-flow across the ocean shorelines and free-slip alongthe ocean shorelines. The free-slip condition is more plau-sible in turbulent flows which have only thin boundarylayers (Schwiderski, 1980). The no-slip condition was alsotested, but the difference was very small. The discreteequations are numerically integrated with time, and timestep is set to 93 sec for the global model and 15.5 sec forthe regional model.

4.2 Tide gauge dataIn order to gain accuracy in the coastal region, the

regional model is assimilated with 219 coastal tide gaugedata around Japan and Korea in addition to the T/P solu-

tions. The data around Japan are provided by the JapanOceanographic Data Center (JODC), and those aroundKorea are kindly provided by Dr. T. Yanagi. The originaldata set of JODC contains harmonic constants from 714tide gauge stations, but the number of reported constitu-ents are different by every station. The number of con-stituents is chosen as a quality index based on an assump-tion that the longer operated stations generally maintainmore constituents and provide high quality data. We se-lected 155 gauges with more than 39 constituents as themost accurate internal gauges and assimilated into thehydrodynamical model. Apart from these gauges we se-lected 80 external gauges with more than 9 (and less than40) constituents which is used for accuracy assessment(TG-C in Subsection 5.1). 64 gauges around Korea arechosen, but only for M2, S2, K1, and O1 constituents whichare all used for assimilation. The geographical distribu-tion of selected gauges around Japan are plotted in Fig.1.

4.3 AssimilationThe T/P solutions and tide gauge data are incorpo-

rated into hydrodynamical model at this assimilationstage. We employ a rather simple assimilation procedureso-called blending method. This method was also intro-duced in tidal modeling and its efficiency was presentedin Kantha (1995) and Matsumoto et al. (1995). The model-predicted tidal height ζMODEL at time step (i) is replacedby a weighted sum of the model prediction and the ob-served T/P tidal height.

ζ ζ ζo mj mjf f′ = ⋅ + −( ) ⋅ ( )T / P MODEL1 12

ff

emjmjc

m

= ( )13

where ζo′ is modified tidal height to be used to calculatethe velocities U and V at next time step (i + 1), ζT/P is thetidal hight predicted by the T/P solution with the normal-ized formal error of em .

The constant values f cmj were determined after sometrial-and-error computations so that the tidal solutionsobtained become smooth in space to some extent. Theemployed values of f c

mj are 0.5 for M2, 0.4 for S2, 0.25for N2, K2, K1, O1, and P1, 0.1 for Q1, 0.05 for 2N2, µ2,ν2, L2, and T2, 0.02 for M1, OO1, and J1. A similar methodis also applied to the tide gauge data in regional tidemodeling, but we assigned a constant nudging factor offmj = 0.95 to the tide gauge data, because published tidalconstants from tide gauges are not usually accompaniedwith their estimation error.

Before stepping forward to final solutions, the fol-

Fig. 1. Location of coastal tide gauge data around Japan usedfor regional assimilation (circles) and for accuracy assess-ment (triangles, TG-C in Subsection 5.1).

572 K. Matsumoto et al.

lowing iterative process is necessary as for SAL tide.(1) It is necessary to know spatial distribution of oceantidal height ζo to calculate the SAL tides which is sug-gested from Eq. (10). The initial tidal field were calcu-lated with the linear approximation ξ = 0.1ζo as the first-guess result from which SAL tides were then estimated.(2) We reprocessed the MGDRB by replacing CSR3.0radial loading values with those predicted by newly esti-mated radial loading charts associated with hn′ in Eq. (10).T/P solutions are also re-estimated from the reprocessedMGDRB data. (3) The linear approximation were replacedin the hydrodynamical model by more precise expressionof Eq. (10) (SAL_NAO.99b) in the final calculation. Theloading tide map is also re-estimated from the final oceantidal solution.

First, we derived global model for major 16 constitu-ents, then they were used to provide open boundary con-dition for regional tidal modeling around Japan. We sepa-rately ran the model for each constituent. Ten-cycle com-putation without assimilation was carried out as spin-uprun and assimilation run was also carried out for ten-cy-cle period which was sufficient to obtain stable results.The amplitude and Greenwich phase values as modeloutput were calculated by Fourier transform of tidal heightfrom the last one cycle of computation.

5. Comparing Models

5.1 Internal comparisonIn order to identify effective aspect of the present

modeling approach in improving the results, we designedseven case runs for the global model and three case runsfor the regional model. At first, bin size of global tidalanalysis is changed to 2 degrees and 1 degree (cases 1and 2) keeping the grid size of numerical model un-changed. Case 3 applies constant nudging factor ( fmj =f c

mj). Cases 4 and 5 exclude FCN resonance effect and

radiational effect out of the tidal analysis, respectively.Case 6 uses linear approximation of SAL tide (SAL_LA),for which β is set to Kantha’s (1995) value of 0.054. Case7 is the standard global model NAO.99b. Case 8 is a re-gional model which uses the SAL_LA and does not as-similate tide gauge data. Case 9 is a regional model whichdoes not assimilate tide gauge data, and case 10 is thestandard regional model NAO.99Jb. These case settingsare summarized in Table 2.

The RMS misfits to three sets of tide gauge data arecalculated for each case run about major eight constitu-ents. The first set of the gauges is a globally distributedisland and sea-bottom tide gauges in open ocean (98 sta-tions) which were selected from the data compiled byC. Le Provost and other members (unpublished data). Thesecond is shallow water (depth < 1000 m) pelagic tidegauges (58 stations) which were selected from Smithson(1992), and the third is the coastal tide gauge data aroundJapan (80 stations) provided by JODC (see Subsection

Case Domain BS NF FCN RAD SAL TG

1 Global 2.0 ED Yes Yes SAL_NAO.99b —2 Global 1.0 ED Yes Yes SAL_NAO.99b —3 Global 0.5 Constant Yes Yes SAL_NAO.99b —4 Global 0.5 ED No Yes SAL_NAO.99b —5 Global 0.5 ED Yes No SAL_NAO.99b —6 Global 0.5 ED Yes Yes SAL_LA —7 Global 0.5 ED Yes Yes SAL_NAO.99b —8 Regional 1/12 ED Yes Yes SAL_LA No9 Regional 1/12 ED Yes Yes SAL_NAO.99b No

10 Regional 1/12 ED Yes Yes SAL_NAO.99b Yes

Table 2. Summary of case settings, in which BS = bin size in degrees used in the altimetric tidal analysis; NF = nudging factor;ED = error dependent; TG = coastal tide gauges around Japan are assimilated or not. See Table 1 for the description of FCN,RAD, SAL, SAL_NAO.99b, and SAL_LA.

Fig. 2. Location of 98 island and sea-bottom tide gauge sta-tions in the open ocean (circles, TG-A), and 58 shallow-water (depth < 1000 m) pelagic tide gauge stations (crosses,TG-B).

Ocean Tide Models from TOPEX/POSEIDON 573

4.2). We hereinafter refer to these three sets of tide gaugesas TG-A, TG-B, and TG-C, respectively. The location ofthe tide gauge stations are plotted in Fig. 2 for TG-A andTG-B, and in Fig. 1 for TG-C (triangles). The RMS mis-fit is defined as

RMS = − −( ) +( )

( )∑1 1

22 142 2

1 2

NA A A AM M T M T T

N

cos/

δ δ

where N is the number of stations, A and δ are amplitudeand Greenwich phase with subscript M and T denotingthe harmonic constants from ocean tide model and tidegauge data, respectively.

Figures 3, 4, and 5 show the results of the compari-son with TG-A, TG-B, and TG-C, respectively, fromwhich we introduce the following six remarks: (1) Thesmaller size of the T/P solution bin results in better accu-racy. This has the largest impact on the accuracy improve-

Fig. 3. RMS misfits to 98 open-ocean tide gauges (TG-A) which compare the accuracy of 7 case runs for the global model. Theindex of each case run is printed at the bottom of the bar. See Table 2 for the description of the case settings.

Fig. 4. Same as Fig. 3, but for 58 shallow-water tide gauges (TG-B).

574 K. Matsumoto et al.

ment of the global model. The relatively larger RMS er-rors of case 1 or 2 for shallow-water gauges (TG-B) thanfor open-ocean gauges (TG-A) suggest the lower perform-ance of 1- or 2-degree grid to capture the short-wave-length feature of ocean tide in shallow seas. (2) The er-ror-dependent nudging factor almost does not changeRMS misfits. This is because the estimation errors of T/Psolutions are generally larger in some specific seas whichare covered by ice in winter, e.g., the Sea of Okhotsk, the

Hudson Bay, and around the Antarctica. Advantage of theerror-dependent nudging factor over the constant nudg-ing factor is not clear in the present comparison in whichfew tide gauges are located at ice-covered seas. (3) K1and P1 constituents are slightly (of the order of 1 mm)improved by including the FCN resonance effect as ex-pected from the fact that their frequencies are relativelyclose to that of the FCN. This is shown by the smallerRMS error of case 7 than that of case 4. (4) The K2 con-

Fig. 5. Same as Fig. 3, but for 80 coastal tide gauges around Japan (TG-C). Also shown are additional comparisons of 3 case runsfor the regional model (cases 8, 9, and 10).

Fig. 6. Vector differences between T/P altimetric tidal solutions and the final assimilated solutions for K1 constituent. Unit is incm.

Ocean Tide Models from TOPEX/POSEIDON 575

stituent is slightly improved by the fine structure of theadmittance which includes radiational anomaly (comparecases 5 and 7). This is not the case for TG-C, but TG-Aand TG-B give the RMS reduction of 0.7 mm and 2.4mm, respectively. (5) For the global model, differencesbetween results using SAL_LA (case 6) andSAL_NAO.99b (case 7) is marginal. However, the use ofSAL_NAO.99b has a more evident effect on the regionalmodel (compare cases 8 and 9). This is because the ratioof the number of T/P-constrained grid to that of total oceangrid is higher for the global model than for the regionalmodel. In other words, if the grid size of a numericalmodel becomes smaller the altimetric constraint of thesame size becomes sparser. And then the quality of thenumerical model, which interpolates tides to the uncon-strained region, becomes more important. (6) The coastaltide gauge data significantly improve the regional model(compare cases 9 and 10). Especially, M2 RMS error re-duction of 1.5 cm is realized by assimilating the coastaltide gauge data.

For the global model we made internal comparisonbetween T/P solutions and final NAO.99b solutions. Fig-ure 6 shows the geographical distribution of the vectordifferences between two solutions for K1 constituent.Large differences are observed in strong current zones,such as the Kuroshio, the Gulf Stream, the Agulhas Cur-rent, or the Antarctic Circumpolar Current. They are alsoevident in the seas which are covered by ice in winterwhere less data are available. This means that the errone-ous T/P solutions are effectively corrected for throughthe assimilation procedure. The discrepancies are largestfor K1 constituent for which global RMS of the vectordifferences is 2.31 cm, while those for M2, O1, and S2 are

1.86 cm, 1.27 cm, and 1.27 cm, respectively. This sug-gests that the estimated admittance at K1 frequency isaffected, to a greater degree, by oceanographic variabil-ity. This is due to the fact that the aliasing period of K1tide (173.2 days) lies in near the period of Ssa which isprimarily weather-driven and has typically wide spectralwidth.

5.2 Comparison with other modelsWe compared the NAO.99b model with the other two

recent ocean tide models. One is CSR4.0 model (improvedversion of Eanes and Bettadpur, 1994) and the other isGOT99.2b model (Ray, 1999). The vector differences forM2 between NAO.99b model and CSR4.0 model is shownin Fig. 7. The differences between NAO.99b model andGOT99.2b model (not shown) has similar spatial patterns.It means that the differences are smaller than 1 cm al-most everywhere in the open ocean, but they are largerthan 5 cm at some specific shallow seas; e.g., the BeringSea, the Sea of Okhotsk, the Yellow Sea, the Europeanshelf, the seas around Indonesia, the seas around Tasma-nia, the Amazon shelf, and Patagonian shelf. They arealso seen in the regions polewards of the T/P limit of 66°where models are necessarily less accurate than elsewhereon the globe.

We employed two kinds of test in order to examinethe accuracy of the ocean tide models. The first test iscomparison with tide gauge data, and the second is acollinear residual reduction test. The results of the firsttest are summarized in Tables 3, 4 and 5 for TG-A, TG-B, and TG-C. The RMS values listed in Table 3 (TG-A)are comparable for all the models, as is expected fromthe fact that the model differences in the open ocean are

Fig. 7. Vector differences between NAO.99b model and CSR4.0 model (improved version of Eanes and Bettadpur, 1994). Con-tours are drawn up to 5 cm with interval of 0.5 cm.

576 K. Matsumoto et al.

relatively small. NAO.99b model shows the best agree-ment with TG-B among the three models. GOT99.2bmodel also performs well with TG-B. This seems becauseGOT99.2b uses, as a background model, the local oceantide model of Lambert et al. (1998) in the Gulf of Maineand the Gulf of St. Lawrence instead of FES.94.1, whileCSR4.0 model does not. The accuracy of an empiricalmodel which solves for correction term to a backgroundmodel depends on the quality of the background modelin shallow waters. The comparison with the coastal tidegauge data (TG-C) shows the significant improvement ofNAO.99Jb model over the other three global models. TheRSS value for main eight constituents is the smallest withNAO.99Jb model (3.024 cm). The possible reasons forthe larger RMS values as to TG-C than TG-A are that (1)

Table 3. RMS misfits to open-ocean tide gauge data (TG-A). Unit is in cm. RSS is the root sum squares of major eight constitu-ents.

Table 4. RMS misfits to shallow water tide gauge data (TG-B). Unit is in cm. RSS is the root sum squares of major eightconstituents.

Model Wave

M2 K1 O1 S2 P1 N2 K2 Q1 RSS

NAO.99b 2.144 1.108 0.951 1.061 0.400 0.966 0.641 0.527 3.083CSR4.0 4.477 1.191 1.309 1.245 0.433 1.094 0.844 0.523 5.205GOT99.2b 2.451 1.138 1.305 1.023 0.382 0.742 0.657 0.491 3.379

Number of stations 58 58 58 58 58 58 58 58

Table 5. RMS misfits to coastal tide gauge data around Japan (TG-C). Unit is in cm. RSS is the root sum squares of major eightconstituents.

model ambiguities are still larger in coastal region, and(2) the model resolution of 1/12° is still coarse for suchlocal comparison. The reason (2) become critical if thetide gauges are subjected to local effects due to the shapeof the bay and do not fully represent the tidal field aroundthe stations.

The second test compares the model performancesin terms of the degree of residual reduction at satellitemeasurement normal points. This test is based on the as-sumption that subtracting tides using a better tide modelshould result in a lower residual variance. The test is car-ried out over the T/P cycles 240–258, of which all themodels compared are independent. The tide predictionsoftware provided by each author is used to output elas-tic ocean tidal height (ocean tide + radial loading tide).

Ocean Tide Models from TOPEX/POSEIDON 577

For NAO.99b model, besides the major 16 constituentsadditional minor 33 constituents are inferred from majorones by interpolating or extrapolating tidal admittance infrequency domain using second order polynomials.

In order to clarify the depth-dependence of residuals,we calculate RMS collinear residual within three regionsof the ocean; the very shallow water region (H < 200 m,area 1), the shallow water region (200 m ≤ H < 1000 m,area 2), and the deep water region (H ≥ 1000 m, area 3).The RMS values are listed in Table 6 which shows theimprovement of NAO.99b model in the shallow waters.NAO.99b model provides the smallest RMS values in area1 of 11.20 cm compared with 15.77 cm for CSR4.0 modeland 13.99 cm for GOT99.2b model. In area 2, NAO.99bmodel gives slightly smaller RMS value of 6.98 cm thanCSR4.0 and GOT99.2b models (7.37 cm). The accuracyin the deep seas are comparable for all the models. How-ever, Table 6 suggests that tidal prediction in continentalshelf region (H < 200 m) still has larger ambiguity (11.20cm) compared with that in deep seas (8.56 cm). The sub-tidal variability and non-linear tides which were omittedin both tidal analysis and the hydrodynamical model maybe the main reason of this large RMS value.

6. Tidal Energetics around JapanAlthough the accuracy of ocean tide models in terms

of tidal elevation is significantly improved due to T/Pprecision altimetry, the issues of tidal energetics still re-main unsolved. The total global rate of energy dissipa-tion for M2 constituent is now well established to be2.5 ± 0.1 TW (1 TW = 1012 W). This value is deducedfrom three different space-geodetic technologies; lunarlaser ranging (Dickey et al., 1994), earth-orbiting satel-lite tracking (Christodoulidis et al., 1988), and satellitealtimetry (Cartwright and Ray, 1991). Although the glo-bal dissipation rate is precisely determined, how andwhere the energy is dissipated remains uncertain. Theplausible candidates of the dissipater are friction in shal-low seas and conversion of barotropic tidal energy tobaroclinic one. The latter occurs mainly in ridge and is-

Model Area Total

H < 200 m 200 m ≤ H < 1000 m H ≥ 1000 marea 1 area 2 area 3

NAO.99b 11.20 6.98 8.56 8.65CSR4.0 15.77 7.37 8.55 8.96GOT99.2b 13.99 7.37 8.65 8.92

Number of normal points 25609 18796 525054 569459

Table 6. RMS collinear residuals. Unit is in cm. H denotes the ocean depth. RMS values are calculated from T/P cycles of 240 to258.

land chain system, and estimates of baroclinic dissipa-tion differ from author by author: for instance, Morozov’s(1995) estimate of global M2 baroclinic dissipation rateis 1.1 TW, but Kantha and Tierney (1997) estimated it as360 GW (1 GW = 109 W). On the other hand, the dissipa-tion by friction is generally dominant in shallow seaswhere tidal currents are stronger than in deep seas as de-scribed by Le Provost and Lyard (1997) who assumedthe bottom friction as the only dissipation mechanism.Such regions include the Yellow Sea and the Sea ofOkhotsk. Our focus here is on showing a preliminary re-sult on the barotropic tidal energetics around Japan fordominant M2 and K1 constituents by using high-resolu-tion NAO.99Jb model which is considered to reproducemore realistic tidal current than the global modelNAO.99b.

Tidal energy flux vector E is expressed (Pugh, 1987)as

E A= −( ) ( )1

215ρ δζ ζgHA uv uvcos δ

where (Aζ, δζ) are the amplitude and the phase of tidalelevation and (Auv, δuv) are the amplitude and the phaseof depth-mean tidal currents of (u, v) = (U/H, V/H) com-ponents. The tidal energy fluxes are shown in Figs. 8 and9 for M2 and K1, respectively.

The characteristic of M2 tidal energy flux aroundJapan shows that the energy is mainly transferred to thesouth-west direction in the Pacific, and in particular, partof it enters into the Sea of Okhotsk and the East ChinaSea where it must be dissipated because of almost no fluxgoes out of the seas. The flux pattern of K1 in the Pacificoffshore is similar to that of M2, but few flux enters ontothe East China Sea and the most of energy seems to bedissipated in the Sea of Okhotsk. There is an energy fluxgyre near to the place (155°E, 47°N) which is probablytrapped by bottom topography. The topography-coherentflux is more evident along 1000 m isobath from (151°E,

578 K. Matsumoto et al.

53°N) to (145°E, 55°N) where the flux turns its directionto the eastward forming V shape of the flux.

The mean rate of energy dissipation due to bottomfriction Db and eddy viscosity De is computed from theexpression

DT

ds dtb b

S

T= − ⋅( ) ( )∫∫∫

116

0ρF u

DT

ds dte e

S

T= − ⋅( ) ( )∫∫∫

117

0ρF u

where Fb = (Fbλ, Fb

θ) is the dissipation vector by bottomfriction, Fe = (Fe

λ, Feθ) is that by eddy viscosity, u =

(u, v), and T is tidal period. The integration of total dissi-pation rate D = Db + De over the NAO.99Jb model do-main gives 280 GW for M2 and 117 GW for K1, respec-tively. The main places where strong dissipation occurs

are the East China Sea and the Yellow Sea region (regionA; 115°E–130°E, 20°N–45°N), and the Sea of Okhotskregion (region B; 135°E–165°E, 45°N–65°N). Listed inTable 7 are the dissipation rates integrated over the threeregions of NAO.99Jb model domain, region A, and re-gion B.

As is expected from Fig. 8, high M2 dissipation ratesare calculated for the regions A and B where the inte-grated values are 155 GW and 54 GW, respectively, mostlyaccounted for by the bottom friction. These values arecompared with estimates by Le Provost and Lyard (1997)of 182 GW and 73 GW which are lager than our esti-mates by 27 GW and 19 GW, respectively. In order tounderstand the geographical distribution of the dissipa-tion, the M2 dissipation rate per unit area by bottom fric-tion in regions A and B are plotted in Figs. 10 and 11.The dissipation occurs over the relatively broad area inregion A reflecting large amplitude of tidal current in this

Region Mean dissipation rate

M2 K1

Db De D Db De D

NAO.99Jb model domain (110°E–165°E, 20°N–65°N) 216 64 280 63 54 117Region A (115°E–130°E, 20°N–45°N) 146 9 155 4 2 6Region B (135°E–165°E, 45°N–65°N) 46 8 54 57 32 89

Fig. 8. M2 ocean tidal energy flux vectors in Wm–1 of cross-section.

Fig. 9. Same as Fig. 8, but for K1 energy flux. Superimposed is1000 m isobath.

Table 7. Mean rate of tidal energy dissipation. Unit is in GW. Db is the dissipation rate by bottom friction, De is the dissipationrate by eddy viscosity, and D is the sum of Db and De.

Ocean Tide Models from TOPEX/POSEIDON 579

Fig. 11. Same as Fig. 10, but in the Sea of Okhotsk region.

region, but strong dissipation rate over 1 Wm–2 is con-centrated on some small areas near coast. Although thedissipation rate in region B is generally smaller than thatin region A, the characteristic of localized dissipation alsoapplies to this region. The dissipation by eddy viscosity

is also concentrated in shallow waters, especially in is-land chain system, such as the Kuril Islands, the Izu-Ogasawara Islands, and Ryukyu Islands. The main sinkof K1 tidal energy is in the Sea of Okhotsk, and the meandissipation rate over the region B is 89 GW which ac-counts for 81% of total dissipation within the NAO.99Jbmodel domain.

However, the above mentioned characteristics of lo-calized dissipation mechanism in the shallow waters maynot be representative of the real ocean, because abarotropic numerical model is unable to estimate tidalenergy conversion into baroclinic tide. The internal tidesignal, wavelength of that is approximately 110 km forsemi-diurnal tide assuming a first mode baroclinic wavespeed of 2.5 ms–1 (Kantha and Tierney, 1997), is also fil-tered out of T/P data by the along-track low-pass filter-ing in Section 2.

Niwa and Hibiya (1999) calculates spatial distribu-tion of internal tidal energy flux converted from barotropicone by using a three-dimensional primitive equation nu-merical model, which suggests the strong generation ofinternal tide over spread area around the continental shelfin the East China Sea and around the Izu-OgasawaraRidge, and some in deep seas. Their three-dimensionalnumerical model gives 58 GW as M2 tidal energy fluxconverted into internal tide within the NAO.99Jb modeldomain (Niwa, personal communication), which is 21%of present barotropic estimate of M2 tidal dissipation byNAO.99Jb. As shown by Ray and Mitchum (1997), sur-face manifestation of internal tides, at least its phase-locked part of the signal, can be detected by T/P altimetry.

Fig. 10. M2 tidal dissipation rate per unit area due to bottomfriction in the East China Sea region. Contours are drawnat 0.01, 0.02, 0.05, 0.1 and 0.2 Wm–2.

Fig. 12. High-pass filtered estimates of M2 tidal amplitudes,plotted along 16 T/P ascending tracks. Background shad-ing corresponds to bathymetry, with darker denoting shal-lower water.

580 K. Matsumoto et al.

In spite they showed the results near Hawaii, in this pa-per, we applied the similar analysis to the region nearJapan. Plotted in Fig. 12 are high-pass filtered estimatesof M2 tidal amplitudes which are deduced from point-by-point tidal analysis for the 16 T/P ascending tracks. Thealtimeter data are not subjected to along-track low-passfiltering before this tidal analysis. The tidal estimates arehighly contaminated by the Kuroshio near latitude of35°N. In the region of latitude between 20°N and 30°N,however, there are clear oscillations with amplitudes of3–4 cm and wavelength of order 150 km. These oscilla-tions imply that internal tides are generated by the bot-tom topography, mainly at the Izu-Ogasawara Ridge andthe continental shelf along off the Ryukyu Islands, propa-gating to the deep ocean. Therefor, we will investigatethe subject of the energy budget of the tidal dissipationin detail through more careful analysis and discussion inorder to fill the gap between the barotropic estimate andthe observational evidence.

7. SummaryWe have developed new global ocean tide model and

loading tide model (NAO.99b model) for 16 major short-period constituents which are developed by assimilatingabout five years of T/P altimeter data into numerical hy-drodynamical model. We also developed a regional high-resolution ocean tide model around Japan (NAO.99Jbmodel) which assimilates coastal tide gauge data as wellas T/P data. The new models have improved the accuracyof ocean tide estimation especially in shallow waters com-pared with the other two existing tide models, CSR4.0model and GOT99.2b model. This has been achieved bythe following methodologies applied to the current study;(1) estimation of altimetric tides in small bins, (2) accu-rate tidal analysis by response method in which fine struc-ture of admittance due to FCN resonance and radiationalanomaly is taken into account, (3) precise estimation ofocean-induced self-attraction/loading effects, and (4) as-similating coastal tide gauge data into NAO.99Jb.

The accuracy of the new ocean tide models has beenexamined using tide gauge data and collinear residualreduction test. NAO.99b shows a comparable agreementwith 98 open-ocean tide gauge data as well as CSR4.0and GOT99.2b. The comparison with 58 shallow watertide gauges, on the other hand, supports the better accu-racy of NAO.99b model in shallow seas. The local com-parison with 80 coastal tide gauge data around Japanshows further improvement by NAO.99Jb. It has been alsoshown that NAO.99b model gives smaller collinearresiduals in shallow waters than CSR4.0 and GOT99.2b.

A preliminary result has been introduced as tobarotropic ocean tidal energy dissipation around Japan.The main sinks of M2 tidal energy are the Yellow Sea—the East China Sea region and the Sea of Okhotsk region

within which ocean tidal energy is dissipated at the meanrate of 155 GW and 54 GW, respectively. The K1 tidalenergy is mainly dissipated in the Sea of Okhotsk at themean rate of 89 GW. The geographical plots of tidal dis-sipation suggest that the dissipation is a highly localizedphenomenon in shallow seas. However T/P detectsbroadly distributed surface manifestation of internal tideeven in deep ocean. More complete description of tidalenergy budget, which includes contribution from radialloading tide, internal tide, the energy converted into shal-low-water constituents, will be continued into our futurework.

AcknowledgementsWe are indebted to Dr. T. Yanagi for providing har-

monic constants from coastal tide gauges around Korea.We are grateful to Dr. Y. Niwa for invaluable discussionson internal tidal dissipation. We are also grateful to twoanonymous reviewers for their critical reading and use-ful comments. We are thankful to NASA JPL for prepar-ing the TOPEX/POSEIDON altimeter data. One of theauthors, Koji Matsumoto, is a research fellow of the Ja-pan Society for the Promotion of Science.

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